Problem 81 - Algebra 1: Concepts and Skills

Question

Use a calculator to evaluate the expression. Round the results to the nearest hundredth. $$ \frac{5 \pm 6 \sqrt{3}}{3} $$

Step-by-Step Solution

Answer

The results of the expression \(\frac{5 \pm 6 \sqrt{3}}{3}\) rounded to the nearest hundredth are \(5.13\) and \(-1.80\).

Step 1: Perform Addition and Subtraction Separately

Break down the expression into two separate calculations since there is a \(\pm\) sign. That gives: \[ \frac{5 + 6 \sqrt{3}}{3} \] and \[ \frac{5 - 6 \sqrt{3}}{3} \].

Step 2: Compute the Square Root

Calculate the square root of 3 first because of the order of operations. Using a calculator, \(\sqrt{3} ≈ 1.73205\).

Step 3: Multiplication

Next, multiply 6 with \(\sqrt{3}\), that gives us: \[6 \times 1.73205 ≈ 10.3923\], then do the addition and subtraction with 5: \[5 + 10.3923 ≈ 15.3923\] and \[5 - 10.3923 ≈ -5.3923\].

Step 4: Division

Finally, divide each result by 3. Using the calculator, \[ \frac{15.3923}{3} ≈ 5.13\] and \[ \frac{-5.3923}{3} ≈ -1.80\]

Step 5: Rounding the Result

Round off the results to the nearest hundredth. The rounded results give \[5.13\] and \[-1.80\]

Key Concepts

Order of Operations

In mathematics, the order of operations is crucial to solve expressions correctly. It defines the steps to follow when more than one operation is involved. For the given exercise, the main operations include square root, addition, subtraction, and division. The correct sequence is often remembered by the acronym PEMDAS, which stands for Parentheses, Exponents (like roots), Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). In the expression \( \frac{5 \pm 6 \sqrt{3}}{3} \), the operations must follow this order:

• Calculate the square root of 3, which is an exponent operation.
• Handle the multiplication of 6 with \( \sqrt{3} \).
• Perform the addition or subtraction as indicated by the \( \pm \) sign.
• Finally, perform the division by 3.

Remember, getting the order of operations right ensures you reach the correct solution.

Calculator Use

Using a calculator efficiently is vital for solving complex expressions, especially where manual calculations can be cumbersome. In this exercise, the calculator was crucial in performing the following steps accurately:

• Finding the square root of 3, which approximately equals 1.73205.
• Multiplying 6 by this square root to get around 10.3923.
• Adding 5 to 10.3923 for one part of the expression and subtracting for the other.
• Finally, dividing these results by 3 to find the two possible outcomes.

Each of these steps can be made easier with a calculator, ensuring precision and reducing the chance for manual calculation errors. Also, familiarize yourself with calculator functions such as the square root button and how to handle decimal points effectively.

Rounding Numbers

Rounding numbers is a method used to simplify results and make them easier to interpret. It involves approximating a number to the nearest specified place value, such as the nearest ten, hundred, or, in this case, the nearest hundredth. When rounding to the nearest hundredth, you should look at the third decimal place. If it is 5 or more, you round up the second decimal place by one; if it's less than 5, you leave the second decimal place as it is. For example, in the division results:

• For 5.1300, the third decimal is 0, so it rounds to 5.13.
• For -1.7960, the third decimal is 6, which rounds up to -1.80.

Rounding helps in making the result neat and manageable without losing significant value. It's an essential skill in both academic exercises and real-world applications.